### Factorization Using Difference of Squares

When you learned how to multiply binomials we talked about two special products.

In this section we’ll learn how to recognize and factor these special products.

**Factor the Difference of Two Squares**

We use the sum and difference formula to factor a difference of two squares. A difference of two squares is any quadratic polynomial in the form

#### Factoring the Difference of Squares

1. Factor the difference of squares:

a)

Rewrite

The difference of squares formula is:

Let’s see how our problem matches with the formula:

The answer is:

We can check to see if this is correct by multiplying

The answer checks out.

**Note:** We could factor this polynomial without recognizing it as a difference of squares. With the methods we learned in the last section we know that a quadratic polynomial factors into the product of two binomials:

We need to find two numbers that multiply to -9 and add to 0 (since there is no

We can factor

b)

Rewrite

c)

Rewrite

2. Factor the difference of squares:

a)

Rewrite

b)

Rewrite

c)

Rewrite

3. Factor the difference of squares:

a)

b)

Rewrite

c)

Rewrite \begin{align*} x^2 y^2 - 1\end{align*} as \begin{align*}(xy)^2 - 1^2\end{align*}. This factors as \begin{align*}(xy + 1)(xy - 1)\end{align*}.

### Examples

Factor the difference of squares:

#### Example 1

\begin{align*}x^4 - 25\end{align*}

Rewrite \begin{align*}x^4 - 25\end{align*} as \begin{align*}(x^2)^2 - 5^2\end{align*}. This factors as \begin{align*}(x^2 + 5)(x^2 - 5)\end{align*}.

#### Example 2

\begin{align*}16x^4 - y^2\end{align*}

Rewrite \begin{align*}16x^4 - y^2\end{align*} as \begin{align*}(4x^2)^2 - y^2\end{align*}. This factors as \begin{align*}(4x^2 + y)(4x^2 - y)\end{align*}.

#### Example 3

\begin{align*}x^2 y^8 - 64z^2\end{align*}

Rewrite \begin{align*}x^2 y^4 - 64z^2\end{align*} as \begin{align*}(xy^2)^2 - (8z)^2\end{align*}. This factors as \begin{align*}(xy^2 + 8z)(xy^2 - 8z)\end{align*}.

### Review

Factor the following differences of squares.

- \begin{align*}x^2 - 4\end{align*}
- \begin{align*}x^2 - 36\end{align*}
- \begin{align*}-x^2 + 100\end{align*}
- \begin{align*}x^2 -400\end{align*}
- \begin{align*}9x^2 - 4\end{align*}
- \begin{align*}25x^2 - 49\end{align*}
- \begin{align*}9a^2 - 25b^2\end{align*}
- \begin{align*}-36x^2 + 25\end{align*}
- \begin{align*}4x^2 - y^2\end{align*}
- \begin{align*}16x^2 - 81y^2\end{align*}

### Review (Answers)

To view the Review answers, open this PDF file and look for section 9.10.